HAL Id: hal-01957113
https://hal.archives-ouvertes.fr/hal-01957113
Submitted on 17 Dec 2018
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Resilience
Yiping Fang, Giovanni Sansavini
To cite this version:
Yiping Fang, Giovanni Sansavini. Optimum Post-Disruption Restoration under Uncertainty for En- hancing Critical Infrastructure Resilience. Reliability Engineering and System Safety, Elsevier, 2018,
�10.1016/j.ress.2018.12.002�. �hal-01957113�
Accepted Manuscript
Optimum Post-Disruption Restoration under Uncertainty for Enhancing Critical Infrastructure Resilience
Yi-Ping Fang , Giovanni Sansavini
PII: S0951-8320(17)31006-2
DOI: https://doi.org/10.1016/j.ress.2018.12.002
Reference: RESS 6323
To appear in: Reliability Engineering and System Safety Received date: 28 August 2017
Revised date: 30 March 2018 Accepted date: 13 December 2018
Please cite this article as: Yi-Ping Fang , Giovanni Sansavini , Optimum Post-Disruption Restoration under Uncertainty for Enhancing Critical Infrastructure Resilience,Reliability Engineering and System Safety(2018), doi:https://doi.org/10.1016/j.ress.2018.12.002
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
1 Highlights
Propose a stochastic programming for infrastructure restoration under uncertainty
A multi-mode component repair model of higher practicality is considered
Propose a tailored Benders decomposition to effectively solve the model
Show the added value of the stochastic model against its deterministic counterpart
ACCEPTED MANUSCRIPT
2
Optimum Post-Disruption Restoration under Uncertainty for Enhancing Critical Infrastructure Resilience
Yi-Ping Fang1 and Giovanni Sansavini2*
1Chaire Systems Science and the Energy Challenge, Fondation Electricité de France (EDF), Laboratoire Génie Industriel, CentraleSupélec, Université Paris-Saclay, 3 Rue Joliot Curie, 91190 Gif- sur-Yvette, France (Email: [email protected])
2Laboratory of Reliability and Risk Engineering, Institute of Energy Technology, ETH Zurich, Leonhardstrasse 21, 8092 Zurich, Switzerland (Email: [email protected])
ABSTRACT
The planning of post-disruption restoration in critical infrastructures often relies on deterministic assumptions such as complete information on resources and known duration of the repair tasks. In fact, the uncertainties faced by restoration activities, e.g. stemming from subjective estimates of resources and costs, are rarely considered. Thus, the solutions obtained by a deterministic approach may be suboptimal or even infeasible under specific realizations of the uncertainties. To bridge this gap, this paper investigates the effects of uncertain repair time and resources on the post-disruption restoration of a critical infrastructure. A two-stage stochastic optimization provides insights for prioritizing the intensity and time allocation of the repair activities with the objective of maximizing system resilience.
The inherent stochasticity is represented using a set of scenarios capturing specific realizations of repair activity durations and available resources, and their probabilities. A multi-mode restoration model is proposed that offers more flexibility than its single-mode counterpart. The restoration framework is applied to the reduced British electric power system, and the results demonstrate the added value of using the stochastic model as opposed to the deterministic model. Particularly, the benefits of the proposed stochastic method increase as the uncertainty associated with the restoration process grows. Finally, decision-making under uncertainty largely impacts the optimum repair modes and schedule.
ACCEPTED MANUSCRIPT
3
Keywords: Critical infrastructures; system resilience; restoration planning; uncertainty; stochastic programming.
1. INTRODUCTION
Critical infrastructures (CIs) such as power grids, telecommunication networks, transportation networks, are the backbone of our society which depends on their seamless operations to provide services and flows of electrical power, information, energy, materials, etc. [1-4] At the same time, disruptions such as malicious attacks, natural hazards, or technical accidents, become inevitable in today’s increasingly complex and risky operating environment [5]. They can significantly impact CIs performance and cause severe economic losses, such as power outages experienced by 2.5 million customers during the 1994 Northridge earthquake in Los Angeles and by 50 million customers during the 2003 North America blackout. Justifiably, then, critical infrastructure protection (CIP) has become a priority for all nations [2, 6-8].
In recent years, lessons learned from catastrophic accidents and the acknowledgment of unknown hazards have drawn the focus of CIP studies towards the concept of resilience. In this regard, a system should not only be reliable, i.e. having an acceptably low failure probability, but also resilient, i.e.
being capable of effectively absorbing, adapting to and rapidly recovering from disruptive events [2, 9-13]. While resilience can be characterized by different system features and attributes [14], it can be effectively enhanced by developing optimum plans for timely restoring the disrupted service after the occurrence of disruptive events. In this context, the main decision is to determine a schedule of tasks for recovering the failed components [15, 16].
In planning the CI restoration, resources are often limited during the post-disruption phase, e.g. during the restoration of transportation networks, repair crews and equipment are usually extremely scarce in the immediate aftermath of an earthquake. Hence, optimization approaches are typically used to facilitate the identification and scheduling of effective restoration strategies for the rapid reestablishment of system functionality while accounting for limited amount of resources. Various studies have been proposed in the literature in the context of post-disruption CI restoration into a
ACCEPTED MANUSCRIPT
4
mathematical programming framework. Bryson et al. [17] applied a mixed integer programming (MIP) approach for selecting a set of recovery subplans giving the greatest benefit to business operation.
Casari and Wilkie [18] discussed restoration when multiple infrastructures, operated by different firms, are involved. Lee et al. [19] focused on a case of network restoration that involves selecting the location of temporary arcs (e.g., shunts) needed to completely reestablish network services over a set of interdependent networks. A MIP model was proposed to minimize the operating costs involved in temporary emergency restoration. Matisziw et al. [20] proposed an MIP model to restore networks where the connectivity between pairs of nodes is the driving performance metric associated with the network. Nurre et al. [21] studied an integrated network design and scheduling problem for the restoration of CI systems. The problem was formulated using integer programming and a dispatch rule-based heuristics is proposed for its efficient solution. They recently provided a comparative study focusing mainly on model complexity and heuristic dispatch rules for this problem [22]. In addition, González et al. [23] proposed a MIP model for optimal infrastructure system restoration considering joint restoration due to the geographical interdependence between multiple CI systems. Ouyang and Wang [24] studied and compared the effectiveness of five strategies for joint restoration of interdependent infrastructures. Here, they applied a Genetic Algorithm (GA) to generate recovery sequences.
The studies above are based on deterministic assumptions such as complete information on the restoration resources and full knowledge of the activity duration. On the other hand, the restoration of infrastructure systems is complicated by the many decisions to be made in a highly uncertain environment exacerbated by the disaster itself, people’s reaction, and limited capability of information gathering. Several factors introduce uncertainty into the parameters of a disaster situation, i.e.
available restoration resources and repair crews, the time duration of the repair activity of failed components and the required resources that should be allocated to it. For example, in post-earthquake power systems restoration, all on-duty personnel is likely to be immediately available in the aftermath of the event, but off-duty personnel may have difficulty reporting if the roads are damaged or they have suffered personal injury. Neglecting these uncertainties may lead to budget and schedule overruns, compromised performance and ineffective resource allocation in the system restoration
ACCEPTED MANUSCRIPT
5
process [25]. Therefore, preparations and recovery plans must be robust with respect to many scenarios, for which the modeling of uncertain parameters is significant [26].
To the best of our knowledge, few studies have tackled uncertainty in post-disruption CI restoration.
Xu et al. [27] has looked at restoring a power network after an earthquake by scheduling inspection, assessment, and repair operations, where the duration of these tasks are assumed to be random variables with known probability distributions. Instead of solving the stochastic program model, the authors used a GA to produce a priority list of repair tasks, which might be suboptimal. Extensive research efforts have been dedicated from the Operation Research (OR) community to the so-called resource-constrained project scheduling problem (RCPSP) under uncertainty [28, 29]. However, existing approaches from the classical RCPSP under uncertainty cannot be directly applied to the CI restoration planning problem (CIRPP) studied in the present paper due to the following challenges: i) there is a precedence network in RCPSP that defines the finish-start precedence relations among the project activities, whereas the relations are not given and thus should be determined in CIRPP; ii) RCPSP is generally based on the minimization of the makespan, i.e. the time to complete the project, whose calculation is trivial, while the evaluation of the objective function in the CIRPP is complicated by system performance quantification through a physical model, e.g. the maximal network flow model [21].
To bridge this research gap, this paper proposes a stochastic programming [30, 31] approach for the post-disruption CI restoration planning problem (CIRPP) which maximizes the expected system resilience over possible realizations of the uncertain restoration parameters. Specifically, the repair tasks of failed components are modeled as multi-mode activities, in which the repairs are performed with variable allocations of resources, and the processing duration is a function of the amount of allocated resources. For example, the total work content of 18 man-hours can be performed by 1 man in 18 hours, 2 men in 9 hours, or by 1.5 men in 12 hours. Multi-mode repair modeling offers more flexibility than its single-mode counterparts [21, 23, 27], and is thus of higher practicality for operators of CI systems. To tackle the computational burden of the stochastic program, a tailored Benders decomposition algorithm is proposed to solve the mixed-integer equivalence of the original model,
ACCEPTED MANUSCRIPT
6
where the mixed-integer equivalent problem is obtained by scenario generation (sampling) and reduction techniques [32].
The post-disruption CI restoration planning problem (CIRPP) studied in this paper is related to classical maintenance repair problems (MRPs) [33-37]. However, those two problems are essentially different in terms of the following aspects: I) The CIRPP focuses only on the restoration stage after a single, large-scale disruption on critical infrastructure systems, i.e., assuming the damages to the system components have occurred. Conversely, MRPs usually cover the whole failure and repair process considering component failures as the main source of uncertainties; II) the CIRPP focuses on the identification and scheduling of effective restoration strategies for the rapid reestablishment of system functionality while accounting for limited amount of repair resources. Conversely, MRPs usually focus on the available strategies for system maintenance and repair, e.g., different choices of the maintenance periods for the system components and of the number of repair teams to keep on site [35]; III) the CIRPP is considered as a planning problem and thus studied in a mathematical programming and optimization framework in the literature. Conversely, MRPs typically represent the failure and repair processes from a statistical point of view, describe the processes through stochastic models, e.g., using failure and recovery rates, and adopt simulation methods like Monte-Carlo [34, 35].
The main contributions of the present study lie in the following aspects: i) a two-stage stochastic programming model is proposed for the post-disruption CIRPP for system resilience, which, to the best knowledge of the authors, is the first effort in the literature to investigate the effects of uncertainty on the post-disruption CI system restoration by OR methods; ii) the modeling of the multi-mode repair of failed components offers more flexibility than its single-mode counterparts, and is thus of higher practicality for operators of CI systems; iii) a tailored Benders decomposition algorithm is proposed to effectively solve the mixed-integer equivalence of the proposed stochastic program model.
The remainder of this paper is organized as follows. Section 2 introduces the multi-mode CI restoration planning problem. Section 3 formulates it as a two-stage stochastic programming model. In Section 4, the proposed solution methodology is presented. Section 5 illustrates the numerical results
ACCEPTED MANUSCRIPT
7
on a test network based on the British electric power transmission system. Finally, concluding remarks and directions for future research are given in Section 6.
2. POST-DISASTER RESTORATION PROBLEM
In this study, CI systems are modeled as networks, in which the movement of commodities corresponds to flows and the service is provided if the flows exceed the desired supply level. The CI network is defined as a collection of nodes and links; commodities are exchanged across nodes along paths in the network. Network flow problems have been extensively studied in the literature and fundamentals can be found in [38]. Mathematically, an infrastructure is represented as an undirected graph comprising a set of nodes (vertices) connected by a set of links (arcs) . The network nodes are classified into supply nodes , transshipment nodes , and demand nodes (
). Each arc is characterized by flow capacity , and each supply node has a supply capacity ̅ in the time period ; each demand node has a demand ̅
in time period describing nominal operations. System performance, i.e. system service level , is measured as a function of the percentage of demand that can be met throughout the system at time .
∑ ̅
∑ ̅ (1)
where is the amount of unsatisfied demand at node at time , and we have ̅ .
The main system components, i.e. supply units, transmission lines, and relay nodes, may be subject to damage stemming from natural disasters or manmade malicious attacks. The proposed restoration planning model aims to reestablish connectivity between supply and demand nodes in a disrupted area by repairing damaged components over a fixed planning horizon, so that the interruption of services at the demand nodes is minimized and system resilience is maximized. Disruptions are modeled by the removal of a subset of arcs, , and a subset of nodes from the network. The arcs and nodes in sets and become non-operational immediately after the disruption, and system performance deteriorates to the minimum level. Successively, system restoration is planned and initiated for ensuring system service by scheduling the individual repair tasks. The planning horizon
ACCEPTED MANUSCRIPT
8
for restoration is divided into discrete time periods, i.e., , and identifies the planning instant. The evolution of system performance versus time, i.e. the restoration curve, typically quantifies the level of system resilience following disruptions [10, 14, 24]. Resilience can be measured by the cumulative system service that is restored during the restoration horizon, normalized by the expected cumulative system functionality supposing that the system has not been affected by disruption during this time period [10, 39]:
∑ [ ]
∑ [ ] (2)
where denotes the targeted system performance if not affected by the disruption, and it is assumed to remain invariant in this study for simplicity. Duration is defined as the timespan necessary to restore the system functionality to the same level as the original system. The restoration horizon evolves in discrete time periods; therefore, Eq. (2) contains the summation operator. Note that is in the range of [ ] since always hold during the restoration process. Then, the system resilience loss is defined as .
The restoration of a damaged component , can be performed by variable allocations of resources , and consequently the repair duration, i.e. the time to repair, is a function of such allocation, i.e. . As exemplified in Figure 1, the repair duration is measured in man-hour units defined as the amount of work performed by the average worker during one hour. The higher the number of crews allocated per hour, the shorter repair time for the damaged component, and shows a saturation point at which further allocating repair crews does not expedite the repair process.
Moreover, there is a minimum amount of resources required for initiating the repair process of a damaged component. The relationship can vary across different components. In practice, this information can be obtained from the historical repair data of the considered CI system.
ACCEPTED MANUSCRIPT
9
Figure 1. Repair time as a function of the number of allocated repair resources per hour.
In this study, the multi-modal restoration of the damaged component is represented by pairs of repair duration and allocated resources, i.e. ̃ , for each discrete repair mode , where is the number of modes in which the repair task of can be implemented. System operators tend to allocate more resources, e.g. repair crews and equipment, to accelerate the restoration of those components which are deemed as critical for the recovery of system functionality. Multiple alternatives usually exist for executing the repair activity based on the combination of the repair modes to which resources are allocated.
The repair time ̃ for the repair mode of component is a stochastic quantity and is modeled by a random variable that may follow diverse probability distributions. Following the most common choice for the probability distribution of activity time [40], the time to repair is Weibull distributed:
̃ { ( )
(3)
where is the shape parameter and is the scale parameter, . For the multi-modal repair activity ̃ , the assignment of external repair resources to a damaged component leads to a reduction of the expected repair time of the component [25]. Hence, the mean time to repair (MTTR) of component can be expressed as a function of the allocated resources :
(4)
where is the gamma function and represents the mean of a Weibull random variable, and can be any non-increasing function of which account for the decreasing repair efficiency
ACCEPTED MANUSCRIPT
10
as more resources are allocated to the repair task. In this study, ( ) , where and are the upper and lower bounds of the expected time to repair for component and is the saturation parameter, reflecting the fact that each extra unit of repair resources allocated to mode has a smaller effect than the previous one due to the decreasing efficiency of recovery activities when the concentration of forces grows. Without loss of generality, any other appropriate relationship can be used to model the saturation of the repair efficiency. Classic parameter estimation techniques such as maximum likelihood and bayesian methods can be applied for calibrating the parameters of the Weibull distribution for the component recovery times, i.e., Eq. (3), and of the relation function, i.e., Eq. (4), based on the historical statistics of the component recovery process..
The total amount of repair resource units that can be allocated to the restoration activities at each time period is denoted by ̃, , which is usually uncertain at the moment of planning the restoration, i.e. at . Therefore, the prediction of ̃ for should be made by the decision maker at and it is usually based on expert judgement and past experience. In this work, the prediction is provided in terms of the minimum, the most-likely and the maximum values of ̃ . Based on the aforementioned representation, the triangular probability distribution is a natural and relatively simple description of the uncertainty associated with the subjective estimation made by the decision maker:
̃ {
(5)
where , , are the minimum, most-likely and maximum values of ̃, respectively, which are estimated by system operators.
In summary, for a disrupted CI system , given (i) a set of repair execution modes for each damaged component , and (ii) a set of uncertainty characterized by activity duration ̃ ̃ and (iii) by total available repair resources ̃ ̃ , the multi-mode CIRPP attempts to determine the optimum combination of repair modes and the associated starting time under each uncertainty realization which minimizes the expected interruption of services at all the demand nodes and maximize the resilience of the system.
ACCEPTED MANUSCRIPT
11
The multi-mode CIRPP involves determining a repair mode for each failed component, independently of uncertainty scenario, and assigning a starting time to each repair activity under each scenario. The decision process in the CIRPP is a stage-wise scheduling of repair activities subject to resources constraints at each time according to selected execution modes. As the (static) mode selection variables are common to all uncertainty scenarios and do not depend on observed values of random parameters ̃ and ̃ , the problem can be viewed as a two-stage stochastic programming with recourse in which the variables related to the repair mode are determined in the first stage and the repair activities are scheduled in the second stage based on the first-stage decisions.
3. TWO-STAGE STOCHASTIC PROGRAMMING FORMULATION
The stochastic multi-mode CIRPP model employs the following notation.
A. Indexes, parameters and sets
An undirected graph comprising a set of nodes connected by a set of links representing the original network system before the disruption.
The sets of supply, transshipment and demand nodes, respectively, The set of identified damaged links after a disruption,
The set of identified damaged nodes after a disruption, The total number of time periods in the restoration planning
The capacity of line
̅ The supply capacity in supply node at time ̅ The demand at node at time
The number of mode for repairing damaged line The number of mode for repairing damaged node
The number of repair resources allocated to a damaged line under mode
The number of repair resources allocated to a damaged node under mode
̃ The time duration expected to repair a damaged line under mode ̃ The time duration expected to repair a failed node under mode
̃ The total amount of repair resource units that can be allocated to restoration activity at time
Index of nodes, Index of lines,
Index of nodes and lines, Index of repair model
Index of the uncertainty scenario which is defined as a realization of all the uncertain
ACCEPTED MANUSCRIPT
12 parameters
B. Decision variables
Whether or not a damaged component is selected to be repaired under mode
Whether or not a damaged component is assigned to repair capacity at time under scenario
Whether or not a failed node or a failed arc is being processed by repair crews at time under scenario
Operational state (functional/failed) of node or arc at time under scenario
The amount of flow passing through arc at time under scenario
The amount of generated flow at node at time under scenario The amount of unsatisfied demand at node at time under scenario
The problem is formulated as a two-stage stochastic programming with recourse. The general formulation of a two-stage stochastic linear program with recourse is [32]:
[ ] s.t.
(6)
where is the cost vector, [ ] is the first-stage decision variable vector, is the coefficient matrix of the first stage variables, is the right hand side vector and is the domain of variable . The function is the second-stage value function (or recourse function) defined as [41]:
{
} (7)
where [ ] is the second-stage decision variable vector and is its domain, is the recourse penalty coefficient, is the coefficient matrix of the first stage variables in the second stage problem’s constraints (called technology matrix), and is the coefficient matrix of the second stage variables in the constraints (called recourse matrix); is a realization of random parameters [ ̃ ̃], indexed by , and is the scenario index set. The second- stage problem in Eq. (7), wherein decisions are made after the uncertainty on ̃ and ̃ is cleared, is referred to as the recourse problem.
Specifically, the stochastic programming model of the CIRPP with the objective of minimizing the expected system resilience loss, i.e. maximizing resilience, is:
ACCEPTED MANUSCRIPT
13
̃ (8)
s.t. ∑ ∑ (9)
∑ ∑ (10)
∑ ∑ ̅ (11)
̅ (12)
̅ ̅ (13)
(14)
∑ (15)
∑ ∑ (16)
∑ (17)
(18)
∑ ∑ ∑ [ ] (19)
∑ ∑ (20)
∑ ∑ (21)
(22)
(23)
(24)
ACCEPTED MANUSCRIPT
14
The system resilience loss under each scenario in the objective function (8) is calculated by Eqs. (1) and (2). Constraints (9)-(14) are the constraints for the operation of general infrastructure network systems, i.e. constraints (9)-(11) guarantee the flow balance at generation, transship and demand nodes, respectively, constraints (12) ensure that the flow generated at a supply node does not exceed its capacity if it is operational, constraints (13) ensure that the amount of flow delivered to a demand node does not exceed the requested demand, and constraints (14) force the arc flows to comply with capacities taking into account that a flow can exist only if the arc and its ending nodes are functional.
Furthermore, constraints (15)-(21) capture the relationships among the restoration scheduling decisions. Constraints (15) ensure that only one repair mode is chosen for each damaged component.
Constraints (16) state that if a damaged component is selected to be repaired, then a repair resource must be assigned to it. Constraints (17) enforce that a damaged component assigned to a repair crew is either under repair or functional, i.e. its restoration is completed. Constraints (18) ensure that restored components do not fail again over the planning horizon. Constraints (19) states that components are immediately available as soon as the repair workforce completes the task. Moreover, constraints (20) ensure that once the restoration of a component is started, it continues at least for the estimated time to repair duration for the chosen repair mode. Constraints (21) limit the use of available repair resources at each time unit. Finally, constraints (22)-(24) describe the binary nature of the decision variables.
The above CIRPP model (8)-(24) is most applicable to single-commodity infrastructure systems which include, for instance, power, water/waste water, natural gas, and supply chain systems. It conveniently extended to multi-commodity systems by adding different indexes to the network flow variables representing different types of commodities, e.g., models in [23], [42] and [43], and by accounting for additional constraints of multi-commodity systems such as the interdependency among the flows of different commodities [38].
4. SOLUTION METHODOLOGY
A. Scenario Generation and Reduction
Owing to the presence of continuous random variables, i.e. Weibull-distributed repair times for each damaged component and triangular-distributed repair resources at each time step, the stochastic
ACCEPTED MANUSCRIPT
15
parameter vector of the proposed model has an infinite support. To obtain a solution to the problem, the stochastic parameter vector needs a discrete representation which restricts the possible outcomes to a finite set. In this study, Latin hypercube sampling [44] is applied to replace via a scenario tree approximation which builds a finite and large set of uncertainty scenarios, , over which a discrete probability distribution is defined. The Latin hypercube sampling guarantees that the whole range of the random variables is spanned. For a sample size of , the Latin hypercube sampling technique selects different values of the random variables by dividing their range into disjoint intervals.
Then, scenarios are built by shuffling and pairing these values; the probability of each scenario is .
The computational tractability of the stochastic optimization model is achieved by reducing the number of scenarios while still preserving the essential features of the original sampling set. In other words, we seek a reduced scenario set that yields an optimum solution whose value is close to the solution of the original optimization problem. Consider the uniform discrete probability distribution over the sampled scenario set , the scenario reduction problem amounts to determining a scenario subset of cardinality and assigning new probabilities to the reduced scenarios such that [32]:
{
∑
∑
∑
}
(25)
where is called Kantorovich distance and is the most common probability distance used in stochastic programming, and represent the probabilities of scenarios and in sets and according to probability distributions and , respectively, and is a nonnegative, continuous, symmetric function, often referred to as cost function and is represented by a norm defined in the scenario space. Further details about Kantorovich distance can be found in [45].
The above problem can be solved through diverse techniques; in this paper, the fast forward selection algorithm is selected because the reduced sets defined by this heuristic algorithm perform well in
ACCEPTED MANUSCRIPT
16
practice [46, 47]. The development and application of the fast forward selection algorithm are detailed in [32] and in [46].
In summary, the infinite support of the stochastic parameter vector of the proposed model is firstly reduced to a sample size of by Latin hypercube sampling and then to a computationally tractable number of scenarios, , using the fast forward selection algorithm. The scenario reduction process implements the fast forward selection method and ensures that the reduced scenario set has the closest probability distance with the original sample space by solving the Kantorovich distance problem (25). Therefore, the reduced scenario set has the theoretical guarantee to be an unbiased representation of the original probability distribution.
B. Tailored Benders Decomposition Strategy
By sampling the scenario tree and reducing it to a tractable number of scenarios, the proposed stochastic programming is converted to its deterministic mixed-integer programming equivalence. As a result of this, a Benders’ decomposition method can be applied to its solution which offers an efficient strategy for large-scale MIP. As opposed to the original problem, the relaxed master problem (RMP) and the Benders’ subproblem (SP) in the Benders’ decomposition are much more tractable.
This section proposes a tailored Benders decomposition algorithm for the effective solution of the CIRPP (8)-(24).
In the classic Benders decomposition framework [48], an original MIP problem is transformed into an RMP and an SP that are iteratively solved and the solution of one problem is given as input to the other, until the optimal solution is achieved. At each iteration, there are two types of cuts that can be generated based on the solution of the SP and then are added to the RMP: i) The first type is optimality cuts that attempt to increase the lower bound of the minimization problem (or the upper bound of the maximization problem), which is obtained from the RMP. ii) The second type of cuts is feasibility cuts, which are used to remove the RMP solutions that are not feasible in the SP. Instead of using only one SP at each iteration, a set of subproblems are formulated and solved, one for each of the scenarios.
Thus, multiple cuts can be generated and added into the RMP at each iteration. showed that, By doing so, the computational performance of the algorithm improves [31]. Given an instance of the binary
ACCEPTED MANUSCRIPT
17
decision vector ̅ [ ̅ ̅ ̅ ̅ ], the SP associated with problem (8)-(24) under scenario is formulated as:
SP( ̅ )
∑ ∑
(26)
s.t. (9)-(14) for one specific scenario and ̅ (27) Since SP( ̅ ) (26)-(27) is a pure linear program, the strong duality theory holds, and its dual problem, called dual subproblem DSP( ̅ ), can be compactly written as:
DSP( ̅ )
̅ (28)
s.t. (29)
where is the right-hand side vector of (27), is the left-hand side coefficient matrix of (27), is the dual variable vector corresponding to constraint (27) and represents the dual feasible region to SP( ̅ ). The set of extreme points of is denoted as , and the set of extreme rays of is denoted as . In the iterative Benders decomposition, the RMP is formulated as follows:
RMP
∑ (30)
s.t.
̅ (31)
̅ (32)
(15)-(24) (33)
where is the probability associated with scenario and is new introduced slack variables; In (31) we have ̅ , for , and (31) defines the set of currently available Benders optimality cuts, and it is obtained via solving the DSP( ̅ ) for all the scenarios in the previous iterations; In (32) we have ̅ , for , and (32) defines the set of currently available Benders feasibility cut. At a given iteration of the Benders decomposition, RMP is solved first to obtain the optimal solutions of ̅ . Then, the optimal solutions of ̅ are used to solve DSP( ̅ ) and generate the
ACCEPTED MANUSCRIPT
18
optimal solutions (or the extreme ray if DSP( ̅ ) is unbounded) of the dual variables ̅ and a new optimality cut in the form of (31) (or feasibility cuts in the form of (32)) to include into the RMP for each scenario . The objective of the RMP provides a lower bound to the optimal solution of the original problem (8)-(24), and the probability-weighted sum of DSP( ̅ ), i.e. ∑ ( ̅ ) ̅ , for all the scenarios provides an upper bound.
For the proposed CIRPP model, it can be observed that:
Proposition 1. For a given binary variable vector ̅ [ ̅ ̅ ̅ ̅ ] which satisfy the constraints (15)-(24), the Benders’ primal subproblem SP( ̅ ) (26)-(27) is always feasible and bounded, i.e. , and the dual subproblem DSP( ̅ ) (28)-(29) is always feasible and bounded and it has at least one optimum solution.
Proposition 1 is easily proven by noticing that the solution characterized by all network flow variables ( ) equal to 0 and by demand shedding equal to the overall demand ( ̅ ) is always feasible. This property circumvents the need to generate feasibility cuts in the decomposition procedure, therefore only optimality cuts are generated and added to the RMP in each iteration, and the convergence of the algorithm is accelerated.
The detailed procedure of the proposed Benders decomposition procedure encompasses the following steps:
Benders decomposition algorithm
Step 0. , iteration counter
Step 1. Solve the RMP (30)(31)(33) to obtain its optimal solution ̅ ̅ , ∑ ̅ Step 2. For each
Solve the DSP ̅ (28)-(29), obtain its optimal solution ̅ and objective value ̅ ̅
End for
∑ ̅ ̅
Step 3. If (where is a predefined tolerance) then stop. Otherwise,
(a) Add a total number of Benders optimality cut ̅ to the RMP (30)(31)(33)
ACCEPTED MANUSCRIPT
19 (b)
Go to Step 1.
The RMP (30)-(33) is a mixed integer nonlinear programming owing to the products of binary variables, i.e. and , in constraints (14) and (21) of the original CIRPP.
According to [49], both nonlinearities can be conveniently replaced by equivalent linear expressions by introducing two sets of binary variables and which satisfy the constraints:
, , (34)
(35)
(36)
. (37) The new binary variables and replace the products of and , respectively, in the original RMP formulation (30)-(33). Therefore, the RMP becomes a mixed-integer linear programming that can be efficiently solved by available off-the-shelf branch-and-cut software, e.g. CPLEX [50].
5. CASE STUDY
The proposed CIRPP is exemplified with reference to a test infrastructure network based on the British high-voltage electrical power transmission system [51, 52]. The original power system includes 400kV and 275kV transmission levels, and the reduced representative network comprises 29 nodes and 99 lines, as shown in Figure 2. The network operation data can be found in [52]. For ease of representation, this analysis considers constant generation capacities and load levels, and only transmission lines can be damaged following a disruption. The number of repair crews is the limited resource allocated to repair damaged lines. Without loss of generality, the shape parameters of the Weibull distribution of the repair time for all components are assumed to be equal to 5, with which the Weibull distribution is generally a symmetric and bell-shaped curve. Each damaged line can be repaired in two modes, i.e. , with one or two allocated repair crews, i.e., or 2, and the associated expected repair times are and , respectively.
ACCEPTED MANUSCRIPT
20
As suggested in [53], the MTTR for transmission line restoration is around 10 hours under the assumption of normal repair workforce.
The considered disruption involves three levels of system failures causing direct damage and loss of functionality to 5, 10, 15 transmission lines ( ). These levels of damage reflect the magnitude of occurred events, e.g. a power blackout in South Australia on October 27, 2016 was caused by severe wind that brought down 6 high-voltage power lines [54]. All the restoration cases are analyzed in a 32-h restoration planning horizon, which is divided into four 8-hours shifts. The chosen planning horizon ensures that all the failed components can be repaired. The parameters of the triangular distribution of the available repair crews during each shift are given in Table 1. The number of available repair crews and the associated uncertainty in a shift are always larger or equal to the same values during earlier shifts. This assumption takes into account the fact that on-duty repair crews are usually available right after the disruption, while all off-duty resources cannot be available immediately; rather their number increase with time [55]. The available number of repair crews within each shift is assumed to be constant. In order to investigate the benefits of using the stochastic model as opposed to the deterministic model that considers only expected values of activity durations, two problems are considered for each disruption case:
ACCEPTED MANUSCRIPT
21
Figure 2. The test network model of the British electric power transmission system.
Table 1. Parameters of the triangular probability distribution of the available repair crews in each shift
i-th shift
1 2.0 3.0 4.0
2 4.0 5.0 6.0
3 6.0 8.0 10.0
4 10.0 12.0 14.0
I) Stochastic program problem (SPP): 1000 independent uncertainty scenarios are generated using the Latin hypercube sampling method [44]. With the use of the fast forward reduction algorithm [32], the number of the uncertainty scenarios is reduced to 5, then, the corresponding equivalent MIPs are solved.
II) Expected value problem (EVP): the deterministic counterpart of the stochastic program is solved, which considers only the expected values of the repair parameter, i.e. expected activity durations and expected available repair crews in each shift.
The aim of solving these two problems is to quantify the value of the stochastic solution (VSS), defined as the difference between the stochastic solution and the expected value solution [31].
Specifically, let [ ] be the optimum solution value of the SPP where represent the objective ̃ in (8), and [ ] be the solution and expected solution value of EVP, then, the difference is referred to as the value of the stochastic solution (VSS) [56]. Furthermore, in this case study, we simply the objective function (8) to ̃ ∑ ∑ by ignoring all the constant items in Eqs. (1) and (2) in order to retain the dimension.
The restoration problems for different disruption cases are solved using the proposed Benders decomposition method, in which the RMP and DSP are solved by CPLEX [50]. Figure 3 shows the objective function values, i.e. the cumulative unsatisfied demand (CUD), for all the six instances, i.e.
SPP and EVP under three disruption cases ( ). The VSS is 0.007 GWh, 0.215 GWh and 0.386 GWh, for disruption scenarios , respectively. Given a household consumes 10.4
ACCEPTED MANUSCRIPT
22
KWh/day electricity in the UK [57], this is equivalent to the daily electricity consumption of around 673, 20’673 and 37’115 household, respectively. This demonstrates an obvious benefit in using the stochastic method over the deterministic one based solely on the expected uncertain parameters.
Figure 3. Objectives values and VVSs under different disruption cases
Figure 4 contrasts the restoration curves of the system, i.e. evolution of the system performance level with time, under the five reduced uncertainty scenarios S1-S5 in the SPP against the restoration curve in the EVP for each of the three disruption cases . In Figure 4, the negative part of the -axis (Restoration Period ) indicates a period before the disruption, Restoration Period marks the occurrence of the disruption, and the remaining periods time the repair process. The performance level in the -axis represents the percentage of the total demand that can be satisfied. As expected, the loss of system performance increases with the number of damaged transmission line, i.e.
0.37% for , 3.54% for , and 5.65% for . The small loss of system performance even in the relatively large disruption is the result of redundant transmission lines that compensate for the offline damaged lines. The behavior of small loss of system functionality can be observed more clearly by combining the system restoration curve and the detailed restoration schedule, e.g. the curve of EVP solution in Figure 4(b) (diamond-dotted line) and its Gantt chart in Figure 5. Repairing only two failed lines, i.e. L22 and L81, leads to a full restoration of the system functionality. In terms of the comparison between SPP and EVP solutions, Figure 4 shows that in each disruption case, i.e., , most of the restoration curves under scenarios S1-S5 in SPP
ACCEPTED MANUSCRIPT
23
outperform the restoration curve in EVP (diamond-solid lines), and it is more evident for large disruptions.
Figure 4. System restoration curves under uncertainty scenarios S1-S5 in SPP and restoration curve in EVP for disruption cases (a) , (b) , (c) .
0 5 10 15 20
0.997 0.998 0.999 1
Restoration Period [h]
Performance Level
S1 S2 S3 S4 S5 EVP (a)
0 5 10 15 20
0.96 0.97 0.98 0.99 1
Restoration Period [h]
Performance Level
S1 S2 S3 S4 S5 EVP (b)
0 5 10 15 20
0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 1.01
Restoration Period [h]
Performance Level
S1 S2 S3 S4 S5 EVP (c)
ACCEPTED MANUSCRIPT
24
Figure 5. Gantt chart of the detailed restoration process of the EVP solutions for disruption case .
Figure 6. Gantt chart of the detailed restoration process of the SSP solutions for the disruption case and for one of the uncertainty scenarios S2.
Figure 6 shows the Gantt chart of the detailed restoration of the SSP solutions for the uncertainty realization S2 for the disruption case . By comparing Figure 6 with Figure 5, one can find that restoration schedules from SSP and EVP are very different. For example, in EVP, lines L22 and L81 are repaired first in mode 1 and 2, respectively, whereas in SSP lines L17 and L81 are repaired first both in mode 2. In other words, consideration of stochasticity in system restoration planning could result in quite different repair modes and time schedule compared to its deterministic counterpart.
Furthermore, we investigate what is the system functionality loss for the solutions of the EVP under each of the five uncertainty scenarios (in terms of the repair duration and available repair crew) other
